Optics for Mathematicagicians

Update: I've added the original gif, linked from Flickr as a movie and it seems to be working a bit better this time.

It's taken me far longer to process this animation file than it has to create the short movie from Mathematica in the first place.

This is all in relation to the talk on atmospheric optics which I'll be giving this week, and is now being advertised on the regional weather website! Not quite sure how that all happened.

I wanted to create a little animation to illustrate some of the properties of a rainbow. In particular I wanted to illustrate Alexander's dark band, the dark region between a primary rainbow and a secondary. To do this you really need to understand the path of light rays going through a water droplet (thankfully water droplets are pretty close to spherical, otherwise the simple trigonometry would become significantly less trivial). I've just done the animation for the ray path for the primary bow, but this illustrates very nicely some of the properties of that bow.

The animation is in two parts. The first part looks at light rays going through a water droplet with two refractions and one reflection. At each refraction point there is a splitting in the colours, though I have only used red, yellow and blue as an example (really for best contrast).

I illustrate these paths for several impact parameters (how far the ray is from that ray which would go straight through the centre of the droplet).

The second part is then to build up the density of light rays so you can see the collective phenomenon (rather than the single path). The final diagram is that of around 100 rays coming in, in an even distribution of impact distances (which is the realistic distribution). One can see several things about this.

The first thing to notice is that none of the light comes out at a greater angle than around 43 degrees to the angle of incidence. However, in the region between about 43 and 41 degrees there is the highest density of rays coming out. The red, which is refracted least, has the highest distribution at around 43 degrees, and the blue, which is refracted most comes out at about 41 degrees. After this, the distribution (between 0 and 41 degrees) is roughly equal for all colours and therefore the light inside the bow is pretty much that of light passing through air (blue).

If you imagine what happens when you populate the sky with such droplets the overall effect is clear. You will see nothing reflected at more than 43 degrees (through single internal reflection), which will give a dark band over the top of the bow - the top of which is red. The secondary bow forms at around 50 degrees and is reversed both in colour and in the region where no light is reflected (there is no light reflected in the region below the secondary bow - giving Alexander's dark band. See here on the atopics website for a great example.

Right, now I've got to work out a good animation for the green flash!